The pricing of an option is a fundamental problem of significant practical importance in today's markets. A wide and growing range of different types of options is available in financial markets around the world. An option is a type of derivative contract whose payoff is derived from the performance of the underlying. An option gives its holder the right but not the obligation to exercise a feature of the contract. For example, an option may provide rights to payments if a temperature in a region reaches a certain level by a certain date. This example option may provide insurance against bad weather. The underlying typically comprises an asset, such as a share, collection of shares or bonds traded on a public market. There are also derivatives based on other assets or instruments, such as commodities, currency values, share price indices and even other derivatives.
The simplest type of a derivative is known as a European “vanilla” option. It gives the buyer the right to buy (call) or sell (put) the underlying asset at a certain fixed time in the future at a given exercise price. American options allow the buyer to buy or sell the asset at the specified price at any time in the future, up to a certain fixed time limit.
While traditional European and American vanilla options specify a fixed exercise price and conditions, path-dependent options have variable exercise prices and conditions, which depend on the behavior of the underlying asset over time. Such options are called exotic. In Asian options, for example, the exercise price is based on an average price of the underlying asset, taken over a certain period prior to exercise. Another example of a path-dependent option is the barrier option, which becomes exercisable only when the price of the underlying instrument rises above or falls below a given threshold.
The challenge for the trader in options and other derivatives is to determine, based on the price and expected behavior of the underlying asset or assets, a price of the derivative justified by the profit that is likely to be made from its future exercise and the potential risk. The challenge is especially difficult at the initial issuance of the derivative. The issuer and potential initial purchaser must determine an accurate price without the benefit of significant data from the market in the derivative.
Even the simplest vanilla options pose a substantial challenge to the derivatives trader, because of the volatile nature of share price values in the stock markets. In deciding whether and when to buy, sell or exercise a given derivative, the trader must assess the factors driving the price of the underlying instrument and the likely variability of the price. These assessments must be factored into a model that can form the basis of a trading strategy that maximizes the profitability of the investment. Models used for this purpose typically view the asset price as a stochastic (random) process.
Various methods are known in the art for determining the stochastic behavior of a process over time, based on a trend function and a variance function. In some simple cases, where the behavior of the underlying asset S can be treated as a scalar log-normal process, Black and Scholes showed how to derive a differential equation whose solution provides the price of a derivative contingent on S. There are many derivatives, however, to which the Black-Scholes analysis is not applicable, such as path-dependent and other exotic options and, more generally, processes that cannot be properly modeled as log-normal. For these more complex analyses, a number of alternatives are known in the art.
Monte Carlo methods can be used to simulate the behavior of the underlying asset and/or derivative over time. Such simulation becomes extremely computation-intensive, however, when high accuracy is required, and small values of the time step must be used. Furthermore, Monte Carlo methods may not work well for pricing American-style path-dependent options. In addition, the use of Monte Carlo methods for actual quantification of risk may be impractical as it may require prices of all derivatives for all levels of the underlying and for all maturities.
Numerical solutions of partial differential equations can be used to evaluate some types of derivatives. These methods, however, are very sensitive to choice of boundary conditions and are likewise demanding of computation resources in multi-dimensional problems. In these problems, the option is dependent upon multiple variables rather than a single variable. One example is a basket option, an option dependent upon the value of a number of stocks or other financial instruments. Similarly, while binomial and trinomial trees are useful, particularly in some path-dependent assessments, they are limited in their ability to deal with multi-dimensional problems. For example, the pricing of basket options with these methods may require a great many nodes of the tree to be evaluated in order to achieve high accuracy.
Dynamic programming techniques provide a method of solving certain types of problems. These techniques may solve a problem by dividing the problem into subproblems and solving the subproblems. The problems may involve variables dependent upon a state. The variables may be indexed; for example, by time period. A simple example of dynamic programming is the determination of the Fibonacci sequence, where F(n+2)=F(n+1)+F(n) and F(0)=F(1)=1. Thus, the problem of determining a value of the sequence for an index decomposes into determining the value of the sequence for two smaller indexed terms. The application of dynamic programming techniques to the valuation of options may have been very limited. The application may be heavily computation-intensive and may not produce estimates of error.